I'm trying to prove that for a stationary and ergodic sequence of martingale differences $X_i$ such that $E X_i = 0$ and $E X_i^2 = \sigma^2 < \infty$ the convergence of $\frac 1n (\sum_{i=1}^{n} X_i)^2$ to $\sigma^2$ is in fact $L_1$ convergence.
I know I have a pointwise convergence of mean of that expression which follows easily from the strong law of large numbers and the fact that those variables are martingale differences. I know it would be true for $ \sum_{i=1}^{n} X_i^2$ because then the uniform integrability would be trivial, but is there any way to prove it for sequence including mixed terms, ie $X_i X_j$? I know I need uniform integrability to obtain that but is it already implied by the assumptions (i.e. stationarity and ergodicity)? If so, what are the tools one can use to prove it?
Thank you very much for your help.
This could be a approach: use a truncation argument and the fourth moments inequality $$E\left(\sum_{j=1}^nD_j\right)^4\leqslant 6 n^2c^4,$$ where $\{D_j\}$ is a sequence of martingales differences such that $|D_j|\leqslant c$ almost surely (for each $j$).
Indeed, let $(X_i,\mathcal F_i)$ be a stationary and ergodic sequence of martingale differences. For a fixed $c$, define $X_i':=X_i\chi_{\{|X_i|\leqslant c\}}-\mathbb E[X_i\chi_{\{|X_i|\leqslant c\}}\mid \mathcal F_{i-1}]$ and $X''_i:=X_i\chi_{\{|X_i|\gt c\}}-\mathbb E[X_i\chi_{\{|X_i|\gt c\}}\mid \mathcal F_{i-1}]$. Then $X_i=X'_i+X''_i$. Denoting $S_n:=\sum_{j=1}^nX_j=S'_n+S''_n$, we have $$\mathbb E[|S_n|^2\chi_{|S_n|\gt 4R}]\leqslant \mathbb E[|S_n'|^2\chi_{\{|S_n'|\gt R\}}]+\mathbb E[|S_n''|^2\chi_{\{|S_n''|\gt R\}}].$$
We have for each positive $R$ that $$\mathbb E[|S_n''|^2\chi_{\{|S_n''|\gt R\}}]\leqslant n\mathbb E[|X_1|^2\chi_{\{|X_1|\gt c\}}]$$ and $$E[|S_n'|^2\chi_{\{|S_n'|\gt R\}}]\leqslant (\mathbb E[S_n'^4])^{1/2}(\mu\{|S_n'|\gt R\})^{1/2}\leqslant 6^{1/2}c^2n^{3/2}\frac c{R}.$$ This gives $$E[|S_n'|^2/n\chi_{\{|S_n'|\gt \sqrt nR\}}]\leqslant \frac Knc^3n^{3/2}\frac 1{\sqrt nR}=\frac{Kc^3}R,$$ where $K$ is independent of $n$, $r$ and $c$.
Plugging these estimates, we get $$\mathbb E[|S_n|^2/n\chi_{|S_n|\gt 4\sqrt nR}]\leqslant K(\mathbb E[|X_1|^2\chi_{\{|X_1|\gt c\}}] +\frac{c^3}R).$$